Proof of a conjecture of N. Konno for the 1D contact process

Abstract

Consider the one-dimensional contact process. About ten years ago, N. Konno stated the conjecture that, for all positive integers $n, m$, the upper invariant measure has the following property: Conditioned on the event that $O$ is infected, the events $\{$All sites $-n, \ldots, -1$ are healthy$\}$ and $\{$All sites $1, \ldots, m$ are healthy$\}$ are negatively correlated.

We prove (a stronger version of) this conjecture, and explain that in some sense it is a dual version of the planar case of one of our results in van den Berg, J., Häggström, O., Kahn, J. (2005), Some conditional correlation inequalities for percolation and related processes, to appear in Random Structures and Algorithms.